<p>Identify the <spanclass="hint_blue">hypothesis</span>, the <spanclass="hint_green">conclusion</span> of the first sentence, and the <spanclass="hint_purple">second sentence</span>.</p>

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<divclass="graphie">

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$("#if-clause").addClass("hint_blue");

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$("#then-clause").addClass("hint_green");

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$("#second-sent").addClass("hint_purple");

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</div>

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</div>

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<p>Does the <spanclass="hint_purple">second sentence</span> refer to the <spanclass="hint_blue">hypothesis</span> of the first sentence, or the <spanclass="hint_green">conclusion</span> of the first sentence?</p>

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<divdata-if="TYPE === 0 || TYPE === 2">

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<p>The <spanclass="hint_purple">second sentence</span> refers to the <spanclass="hint_blue">hypothesis</span> of the first sentence, because they both talk about whether or not <var>IMPLICATION[0]</var>.</p>

62

<p>Does the <spanclass="hint_purple">second sentence</span> state the <spanclass="hint_blue">hypothesis</span>, or the opposite of the <spanclass="hint_blue">hypothesis</span>?</p>

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</div>

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<divdata-else>

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<p>The <spanclass="hint_purple">second sentence</span> refers to the <spanclass="hint_green">conclusion</span> of the first sentence, because they both talk about whether or not <var>IMPLICATION[1]</var>.</p>

66

<p>Does the <spanclass="hint_purple">second sentence</span> state the <spanclass="hint_green">conclusion</span>, or the opposite of the <spanclass="hint_green">conclusion</span>?</p>

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</div>

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<divdata-if="TYPE === 0"data-unwrap>

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<p>The <spanclass="hint_purple">second sentence</span> states the <spanclass="hint_blue">hypothesis</span> of the first sentence.</p>

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<p>Because the <spanclass="hint_purple">second sentence</span> states the <spanclass="hint_blue">hypothesis</span> of the first sentence, the <spanclass="hint_purple">second sentence</span> implies the first sentence.</p>

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<p>Since we are implying the original statement, we can conclude the <spanclass="hint_green">conclusion</span> of the first statement.</p>

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</div>

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<divdata-if="TYPE === 1"data-unwrap>

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<p>The <spanclass="hint_purple">second sentence</span> states the <spanclass="hint_green">conclusion</span> of the first sentence.</p>

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<p>Because the <spanclass="hint_purple">second sentence</span> states the <spanclass="hint_green">conclusion</span> of the first sentence, the <spanclass="hint_purple">second sentence</span> implies the converse of the first sentence.</p>

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<p>Converses are not logically equivalent to their original statements, so we cannot form a logical conclusion.</p>

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</div>

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<divdata-if="TYPE === 2"data-unwrap>

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<p>The <spanclass="hint_purple">second sentence</span> states the opposite of the <spanclass="hint_blue">hypothesis</span> of the first sentence.</p>

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<p>Because the <spanclass="hint_purple">second sentence</span> states the opposite of the <spanclass="hint_blue">hypothesis</span> of the first sentence, the <spanclass="hint_purple">second sentence</span> implies the inverse of the first sentence.</p>

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<p>Inverses are not logically equivalent to their original statements, so we cannot form a logical conclusion.</p>

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</div>

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<divdata-if="TYPE === 3"data-unwrap>

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<p>Because the <spanclass="hint_purple">second sentence</span> states the opposite of the <spanclass="hint_green">conclusion</span> of the first sentence, the <spanclass="hint_purple">second sentence</span> implies the contrapositive of the first sentence.</p>

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<p>Since the contrapositive is implied by the first sentence, the <spanclass="hint_purple">second sentence</span> implies the opposite of the <spanclass="hint_blue">hypothesis</span>.</p>

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</div>

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<p>Thus, the answer is "<var>SOLUTION</var>"</p>

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</div>

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</div>

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<divid="syllogism">

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<divclass="vars">

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<varid="QUESTIONS">

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[[true, true, ["a figure is a square", "it is a rectangle", "a figure is a rectangle", "it has four right angles"]],

<pclass="problem">Use the given information to make a logical conclusion, if possible.</p>

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<pclass="question">If <spanid="hyp_a"><var>HYP_A</var></span>, then <spanid="conc_a"><var>CONC_A</var></span>. If <spanid="hyp_b"><var>HYP_B</var></span>, then <spanid="conc_b"><var>CONC_B</var></span>.</p>

<p>Identify the <spanclass="hint_blue">first hypothesis</span>, the <spanclass="hint_green">first conclusion</span>, the <spanclass="hint_red">second hypothesis</span>, and the <spanclass="hint_purple">second conclusion</span>.</p>

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<divclass="graphie">

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$( "#hyp_a" ).addClass( "hint_blue" );

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$( "#conc_a" ).addClass( "hint_green" );

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$( "#hyp_b" ).addClass( "hint_red" );

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$( "#conc_b" ).addClass( "hint_purple" );

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</div>

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</div>

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<div>

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<p>Do the two sentences come in the form "If <spanclass="hint_blue">P</span>, then <spanclass="hint_green">Q</span>. If <spanclass="hint_red">Q</span>, then <spanclass="hint_purple">R</span>", where <spanclass="hint_green">first conclusion</span> and <spanclass="hint_red">second hypothesis</span> are the same?</p>

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<p>In other words, do the sentences look like <code>\blue{P}\implies \green{Q}</code>. <code>\red{Q}\implies \purple{R}</code>?</p>

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</div>

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<divdata-if="CONC_POSSIBLE"data-unwrap>

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<p>Yes. Because the middle two statements both say <spanclass="hint_green"> <var>HYP_B</var></span>, we can chain the statements together: <code>\blue{P}\implies\green{Q}\implies\purple{R}</code> or <spanclass="hint_blue">"<var>HYP_A</var>"</span><code>\implies</code><spanclass="hint_green">"<var>CONC_A</var>"</span><code>\implies</code><spanclass="hint_purple">"<var>CONC_B</var>"</span>.</p>

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<p>We can now remove the middle statement, and arrive at the conclusion <spanclass="hint_blue">"<var>HYP_A</var>"</span><code>\implies</code><spanclass="hint_purple">"<var>CONC_B</var>"</span>. So, the answer is "If <spanclass="hint_blue"> <var>HYP_A</var></span>, then <spanclass="hint_purple"> <var>CONC_B</var></span>."</p>